A REMARK ON ASSOCIATED PRIMES IN THE COHOMOLOGY ALGEBRA OF A FINITE GROUP (Cohomology Theory of Finite Groups and Related Topics)

A REMARK ON ASSOCIATED PRIMES

IN THE COHOMOLOGY ALGEBRA OF A FINITE GROUP

TETSURO OKUYAMA

奥山 哲郎

HOKKAIDO UNIVERSITY OF EDUCATION, ASAHIKAWA CAMPUS

北海道教育大学・旭川校

Introduction

Let $G$ be a finite group and _{$H^{*}(G, k)$} be the cohomology algebra of $G$

over an

al-gebraically closed field $k$ of characteristic $p>0$. It is well known that _{$H^{*}(G, k)$} is

a

commutative noetherian graded k-algeba. In my talk we discussed properties of the

as-sociated primes in $H^{*}(G, k)$ and gave a slight generalizaton of

a

result of Green [7].

Theorem 0.1.

_If

$\mathfrak{p}$ is

an

associated prime in $H^{*}(G, k)$ with $\dim H^{*}(G, k)/\mathfrak{p}=s$, then

there exists an elementary abelian p-subgroup $E$

of

rank $s$

of

$G$ such that the depth

of

$H^{*}(C_{G}(E), k)$ is $s$ and $\mathfrak{p}={\rm Res}_{G,E}^{-1}(\sqrt{0})$.

Conversely, let $E$ be an elementary abelian p-subgroup $E$

of

rank $s$

of

$G$ and assume

that the depth

_of

$H^{*}(C_{G}(E), k)$ is $s$. Then ${\rm Res}_{G,E}^{-1}(\sqrt{0})$ is

an

associatedprime in _{$H^{*}(G, k)$}.

The first assertion wasknown to be true using the notion of the SteenrodAlgebra [4].We

shall give a different proof using arguments by Carlson [3]. And the second assertion is

known to be true if $C_{G}(E)=G$ by Green [7].

1. The Rank-Restriction Condition

Let $r$ be thep-rank of $G$. Let $A_{s}=\mathcal{A}_{s}(G)$ be the set of elementary abelian p-subgroups

of $G(1\leqq s\leqq r)$ and set $\mathcal{H}_{s}=\mathcal{H}_{s}(G)=\{C_{G}(E) ; E\in A_{s} \}$. And for $1\leqq s\leqq r$,

let $\mathcal{K}_{s}=\mathcal{K}_{s}(G)$ be the family of subgroups $K$ of $G$ such that the Sylow p-subgroups of $C_{G}(K)$

are

not conjugate to

a

subgroup ofanyof the groups in $\mathcal{H}_{s}$. (Ifsuch

a

subgroup $K$

does not exists, then we set $\mathcal{K}_{s}=\{1\}.)$ Notice that any elementary abelian p-subgroup

of $K\in \mathcal{K}_{s}$ have rank at most $s-1$ . For

a

nonempty family $\mathcal{P}$ of subgroups of $G$, set

${\rm Im} Tr_{P,G}=\sum_{P\in P}{\rm Im}$ Tr_$P,G$ and KerRes$G, \mathcal{P}=\bigcap_{P\in P}$KerRes_$G,P$

.

Then by

a

theorem of

Benson (Theoreml.1 [1]) $\sqrt{{\rm Im} Tk_{\mathcal{H}_{s}G}}=\sqrt{Ker{\rm Res}_{G\mathcal{K}_{s}}}$.

Let $\zeta_{1},$

$\cdots,$$\zeta_{r}$ be a homogeneous system of parameters in $H^{*}(G, k)$ satisfying the

rank-restriction condition. Then for $1\leqq s\leqq r,$ ${\rm Res}_{G,E}(\zeta_{S})=0$ for any $E\in \mathcal{A}_{s-1}$,

we

see

that $\zeta_{s}\in\sqrt{Ker{\rm Res}_{G\mathcal{K}_{s}}}=\sqrt{{\rm Im} T\}_{?i_{s}G}}$. Thus replacing $\zeta_{s}$’s by a suitable p-power, we may

assume

that $(_{s}\in{\rm Im} Tk_{\mathcal{H}_{s},G}$ for each $1\leqq s\leqq r$

.

Thus we always have a homogeneous

sys-tem of parameters (1,$\cdots,$ $(_{r}$ in $H^{*}(G, k)$ satisfying the following conditions. See Lemma

8.5 [2] for the condition 3 and Lemma 3.4 and Theorem 9.6 [2] for the condition 4.

Condition 1.1.

(1) $\zeta_{1},$

(2) $\zeta_{s}\in{\rm Im} Tr_{\mathcal{H}_{s},G}$

(3) For each $s$ and $E\in \mathcal{A}_{s}$, the restrictions

of

$\zeta_{1},$

$\cdots,$ $\zeta_{s}$ to $H^{*}(E, k)$

form

a

homo-geneous system

_of

parameters (and $(_{s+1},$ _$\cdots,$ $(_{r}$ restrict to $0$). In particular, their

restrictions to $H^{*}(C_{G}(E), k)$

form

a regular sequence

for

$H^{*}(C_{G}(E), k)$.

(4)

_If

$H^{*}(G, k)$ has depth $s$, then $\zeta_{1},$ $\cdots$ , $\zeta_{s}$

form

a regular sequence.

(5) $\dim H^{*}(G, k)/((s+1, \cdots, \zeta_{r})$ is $s$.

Lemma 1.2 (Carlson [3]). Let $0\neq\eta\in H^{*}(G, k)$ be a homogeneous element.

Assume $\dim H^{*}(G, k)/Ann_{H^{*}(G,k)}(\eta)=s$. Then the following statements hold.

(1)

_If

$t>0$, then

_for

any $H\in \mathcal{H}_{t},$ ${\rm Res}_{G,H}(\eta)=0$.

(2) For

some

$H\in \mathcal{H}_{s},$ ${\rm Res}_{G,H}(\eta)\neq 0$.

Proof.

Set $a=Ann_{H^{*}(G,k)}(\eta)$.

1. Suppose that ${\rm Res}_{G,H}(\eta)\neq 0$for

some

$H\in \mathcal{H}_{t}$ with$t>0$ andset $b=$ Ann_{$H^{e}(G,k)({\rm Res}_{G,H}(\eta))$}

.

Then

as

${\rm Res}_{G,H}^{-1}(b)\subset a$,

$s=\dim H^{*}(G, k)/a\geqq\dim H^{*}(G, k)/{\rm Res}_{G,H}^{-1}(b)=\dim H^{*}(H, k)/b\geqq$depth $H^{*}(H, k)$

$H$ has

a

central elementary abelian p-subgroup of rank $t$ and depth_{$H^{*}(H, k)\geqq t$} by

a

theorem of Duflot [5]. Thus we have a contradiction.

2. Suppose that ${\rm Res}_{G,H}(\eta)=0$ for all $H\in \mathcal{H}_{s}$. Then by the statement 1, _{${\rm Res}_{G,H}(\eta)=0$}

for all $H\in \mathcal{H}_{t}$ with $t\geqq s$. Thus _{$Ann_{H^{*}(G,k)}(\eta)$} contains $(_{s},$ _$\cdots,$$\zeta_{r}$ and

$s=\dim H^{*}(G, k)/$Ann$H^{*}(G,k)(\eta)\leqq\dim H^{*}(G, k)/((S’\cdots, \zeta_{r})\leqq s-1$

This is

a contradiction.

$\square$

2. Results of D.J. Green

In this section we shall prove some results on cohomology algebra of a finite group

having

a

normal elementary abelian p-subgroup. A discussion here depends heavily

on

investigations by D.J.Green [7].

2.1. In this section let $C$ be a finite group with a central elementary abelian _p-subgroup

$E$ of order _$p^{s}(s>0)$ and

assume

that ap-rank of $C$ is larger than $s$. Set $\overline{C}=C/E$ and

for a subgroup $E\subset H\subset C$, set $\overline{H}=H/E\subset\overline{C}$

.

Set $\mathcal{A}=\mathcal{A}_{1}(E)$ be the set of elementary

abelian p-subgroups $F$ of $C$ containing $E$ with $[F : E]=p$ and _{$\mathcal{H}=\{C_{C}(F);F\in \mathcal{A}\}$}.

And let $\mathcal{K}$ be the family of

subgroups $K\subset C$ such that the Sylow p-subgroups of _{$C_{C}(K)$}

are not conjugate to a subgroup of any of the groups in $\mathcal{H}$ (If $\mathcal{H}$ has a subgroup $H$ with

p’-index in $C$, then weset _{$\mathcal{K}=\{1\}$}

as

before). Notice that if$K\in \mathcal{K}$, then anyelementary

abelian p-subgroup of $K$ is contained in $E$.

Let$\underline{F}_{1},$ $\cdots F_{k}-$, be a complete set of representatives of the C-conjugacy classes in $\mathcal{A}$.

Then $F_{i}\subset C(1\leqq i\leqq)$

are

not $\overline{C}$

-conjugate. let $\sigma_{i}’\in H^{2}(\overline{F_{i}}, GF(p))\subset H^{2}(\overline{F_{i}}, k)$ be a

nonzero

fixed element and set $\sigma_{i}=(\sigma_{i}’)^{p-1}$. Then using the Evens’ norm map, there can

be constructed a homogeneous element $\lambda\in H^{*}(\overline{C}, k)$ such that ${\rm Res}_{\overline{C},\overline{F_{l}}}(\lambda)$ is

a

p-power

of $\sigma_{i}$ for each $i$ (see a discussion in Section 7.1 [7]). Now set _{$\kappa=Inf_{\overline{C},C}(\lambda)\in H^{*}(C, k)$}. $\kappa$ is a primitive element. We know that _{${\rm Res}_{C,K}(\kappa)=0$} for any _{$K\in \mathcal{K}$} and therefore,

replacing $\kappa$ by

some

power, we may

assume

that

$\kappa\in{\rm Im} H_{\mathcal{H},C}$ (see a proofof Lemma 2.6

Proposition 2.1. The homogeneous element $\kappa\in H^{*}(C, k)$ given above

satisfies

the

fol-lowing properties.

(1) $\kappa\in{\rm Im}$Tr

$H,C$.

(2) ${\rm Res}_{C,H}(\kappa)\in H^{*}(H, k)$ is a regular element

for

every $H\in \mathcal{H}$.

(3) Ann$H^{*}(C,k)(\kappa)=Ker{\rm Res}_{C,H}$

(4) Suppose that $\xi_{1},$

$\cdots,$$\xi_{t}$ is a sequence

of

homogeneous elements

of

$H^{*}(C, k)$ whose

restrictions

_form

a regular sequence in $H^{*}(E, k)$. Then (a) $\xi_{1},$

$\cdots,$$\xi_{t}$ is a regular sequence

for

$H^{*}(C, k)/H^{*}(C, k)\cdot\kappa$. In particular, as a

$k[\xi_{1}, \cdots, \xi_{t}]$-module, $H^{*}(C, k)/H^{*}(C, k)\cdot\kappa$ is

free.

(b) KerRes$C,\mathcal{H}$ is

a

free

$k[\xi_{1}, \cdots, \xi_{t}]$-module

(5)

_If

the depth

_of

$H^{*}(C, k)$ is $s$, then $\kappa$ is

a

zero

divisor in $H^{*}(C, k)$. In particular,

KerRes$C,\mathcal{H}\neq 0$.

Proof.

All the statements inthe proposition except the assertion $(b)$ in thestatement 4

are

included in [7]. We shall prove the stetement 4, $(b)$. Set $R=k[\xi_{1}, \cdots, \xi_{t}]$

.

Then _{$H^{*}(C, k)$}

is

a

free R-module by a result of Duflot [5]. We have the following exact sequence of

R-modules

$0arrow$ Ann$H^{*}(C,k)(\kappa)arrow H^{*}(C, k)arrow\kappa H^{*}(C, k)arrow H^{*}(C, k)/H^{*}(C, k)\kappaarrow 0$

and Ann$H^{*}(C,k)(\kappa)=Ker{\rm Res}_{C,H}$ . Thus by the assertion $(a),$ $4$, it follows that _{$Ker{\rm Res}_{C,H}$}

is a projective R-module. So the result follows because a projective R-module is a free

R-module (see, for example, Theorem 2.5 [8]). $\square$

2.2. In this section let $N$ be a finite group with normal elementary abelian subgroup $E$

of rank $s$ and set $C=C_{N}(E)$. We

use

notations in the previous section for $C$. We shall

prove the following proposition.

Proposition 2.2. Assume that the depth

_of

$H^{*}(C, k)$ is $s$. Then

(1) As a $kN/C$_-module $Ker{\rm Res}_{C,\mathcal{H}}$ contains a regular module $kN/C$.

(2) $Ker{\rm Res}_{C,\mathcal{H}}$ contains a

nonzero

N-invariant element.

Set $E=\langle a_{1},$ _{$\cdots,$ $a_{s}\},$ $\alpha_{i}\in H^{1}(E, k)=Hom(E, k)$} be the element dual to

$a_{i}$ and $\beta_{i}=$ $\beta(\alpha_{i})$, where $\beta$ is the Bockstein map. We have a polynomial subalgebra $k[\beta_{1}, \cdots, \beta_{s}]$ in

$H^{*}(E, k)$. Using Evens’ norm map, we obtain homogeneous elements $\xi_{1},$ $\cdots\xi_{s}\in H^{*}(C, k)$

such that ${\rm Res}_{C,E}(\xi_{i})=\beta_{i}^{p^{a}}$ for

some

p-power_$p^{a}$. The subalgebra $k[\beta_{1}, \cdots, \beta_{s}]\subset H^{*}(E, k)$

is N-stable, but thesubalgebra $k[\xi_{1}, \cdots, \xi_{s}]\subset H^{*}(C, k)$ may not be N-stable. For_{$g\in N$},

let $\beta_{i}^{g}=\sum_{j=1}^{s}\lambda_{ij}\beta_{j},$ $\lambda_{ij}\in GF(p)$ and consider the element $\xi=\xi_{i}^{g}-(\sum_{j=1}^{s}\lambda_{ij}\xi_{j})$. Then ${\rm Res}_{C,E}(\xi)=0$ and therefore ${\rm Res}_{C,K}(\xi)$ is nilpotent for any $K\in \mathcal{K}$ because elementary

abelian p-subgroups of such $K$

are

contained in $E$

.

Hence by a theorem of Benson

(The-orem

1.1 [1]$)$

some

p-power of $\xi$ is contained in _{${\rm Im} Tr_{H,C}$}. Replacing the $\xi_{i}’s$ by suitable

p-powers, we have proved the following.

Lemma 2.3. There exist homogeneous elements$\xi_{i}(1\leqq i\leqq s)$ and ap-power$p^{b}$ satisfying

the following.

(2) $k[\xi_{1}, \cdots, \xi_{s}]+{\rm Im} Tr_{\mathcal{H},C}$ is N-stable and$k[\xi_{1}, \cdots, \xi_{s}]+{\rm Im} Tr_{\mathcal{H},C}/{\rm Im} Tr_{\mathcal{H},C}\cong k[\beta_{1}^{p^{b}}, \cdots, \beta_{1}^{p^{b}}]$

as

N-modules.

Proof of

Proposition 2.2

Now

we

shall prove the proposition. First

we

prove the statement 1. Set $R_{0}=$

$k[\beta_{1}^{p^{b}}, \cdots , \beta_{s}^{p^{b}}]$ and $R=k[\xi_{1}, \cdots , \xi_{s}]$. Then it is well known that the

sequence

$\xi_{1},$_$\cdots,$$\xi_{s}$

is

a

regular sequence for $H^{*}(C, k)$ and $H^{*}(C, k)$ is

a

free R-module (see [5]).

By Proposition 2.1, $Ker{\rm Res}_{C,7\{}\neq 0$. Let $n$ be the least integer such that $V=$

KerRes$C,H\cap H^{n}(C, k)\neq 0$. Then $V$ is N-stable and any k-basis of$V$ canbe extendedto a

set of R-free generators of Ker Res$C,\mathcal{H}$. Then by Lemma 1.2, Proposition 2.1 and the fact

that ${\rm Im} Tr_{7\{,C}$ annihilates Ker Res$C,H$, it follows that $V\cdot R$ is N-stable and $V\cdot R\cong V\otimes_{k}R_{0}$

as

N-modules.

It is known that $R_{0}$ contains $kN/C$

as

a

$kN/C$-module (see [9]).

Therefore

the statement (1) follows.

The statement 2 is

an

easy consequence of the statement 1.

3. Proof of Theorem 0.1

First we shall show that the first statement of the theorem hold. Let $\mathfrak{p}$ be

an

associated

prime in $H^{*}(G, k)$ with $\dim H^{*}(G, k)/\mathfrak{p}=s$ and take

a

homogeneous element $0\neq\eta\in$

$H^{*}(G, k)$ such that Ann$H^{*}(G,k)(\eta)=a$. Then by Lemma 1.2, there exists $E\in \mathcal{A}_{s}$ such

that ${\rm Res}_{G,C_{G}(E)}(\eta)\neq 0$. Set $C=C_{G}(E),$ $\eta_{0}={\rm Res}_{GC)}(\eta)$ and

we

shall

use

the notations in Section 2. Again by Lemma 1.2, $\eta_{0}\in$ KerRes_$C,?t$.

Let $\alpha\in H^{*}(C, k)$ such that ${\rm Res}_{C,E}(\alpha)=0$. Then $\alpha\in\sqrt{Ker{\rm Res}_{C\mathcal{K}}}$by

an

argument in

Section

1 and therefore $\alpha\in\sqrt{{\rm Im} Tr_{HC}}$. Thus $\alpha\in\sqrt{Ann_{H^{*}(Ck)}(\eta_{0})}$.

By

a

result of Benson in [1] there exists a homogeneous element $\tau\in H^{*}(C, k)$ such that ${\rm Res}_{N,C}$Tr$C,N(\tau)$ is a regular element in $H^{*}(C, k)$ and restricts to zero on every subgroup

$A$ of $C$ with $A\not\supset E$, where $N=N_{G}(E)$. Set $\sigma=$ Tr$c,c(\tau)$ and consider the element $\alpha=\eta\sigma$. We shall show the following equality hold.

${\rm Res}_{G,C}(\alpha)=\eta_{0}{\rm Res}_{N,C}Tr_{C,N}(\tau)$

As $\alpha=\eta$Tr$c,c(\tau)=Tr_{C,G}({\rm Res}_{G,C}(\eta)\tau)=$ Tr$c,c(\eta_{0}\tau)$, the Mackey double coset formula shows that

${\rm Res}_{G,C}( \alpha)=\sum_{g\in C\backslash G/C}$Tr$c 9\cap c,c{\rm Res}_{C^{g},C^{g}\cap C}((r_{0}\tau)^{g})=\sum_{g\in C\backslash G/C}$Tr

$c9\cap c,c{\rm Res}_{C^{g},C^{g}\cap C}(\eta_{0}\tau^{q})$

Suppose that $C^{-1}\not\supset E$. Then

as

_{${\rm Res}_{C,C\cap C^{g^{-1}}}(\tau)=0$}, it follows that${\rm Res}_{C,C\cap C^{g^{-1}}}(\eta_{0}\tau)=0$

and therefore ${\rm Res}_{C^{g},C^{g}\cap C}(\eta\tau^{g})=0$. If $E\subset C^{g^{-1}}$ and $E\neq E^{g^{-1}}$, then $F=EE^{g}$

‘1

is

elementary abelian, $F\supsetneq E$ and $C\cap C^{g^{-1}}=C_{G}(F)$. So ${\rm Res}_{C,C\cap C^{g^{-1}}}(\eta_{0})=0$ and therefore ${\rm Res}_{C^{g},C9\cap C}(\eta\tau^{g})=0$. Thus

we

have the desired equality. a $\neq 0$ and therefore $\mathfrak{p}=$

Ann$H^{*}(G,k)(\eta)=$ Ann$H^{*}(G,k)(\alpha)$. In these notations, we shall show that for $\rho\in H^{*}(G, k)$, $\rho\alpha=0$ if and only if ${\rm Res}_{G,C}(\rho)\eta_{0}=0$.

Assume that ${\rm Res}_{G,C}(\rho)\eta_{0}=0$, then ${\rm Res}_{G,C}(\rho)\eta_{0}\tau=0$ and $\rho\alpha=$ Tr$c,c({\rm Res}_{G,C}(\rho)\eta_{0}\tau)=$ $0$. Thus we have $\mathfrak{p}={\rm Res}_{G,C}^{-1}($Ann$H^{*}(C,k)(\eta_{0}))$.

As ${\rm Res}_{\overline{C}^{1}E}(\sqrt{0})\subset\sqrt{Ann_{H^{*}(Ck)}(\eta_{0})}$, we have

${\rm Res}_{G,E}^{-1}(\sqrt{0})\subset{\rm Res}_{G,C}^{-1}($Ann$H^{*}(C,k)(rlo))=\mathfrak{p}$

As ${\rm Res}_{G,E}^{-1}(\sqrt{0})$ is

a

prime ideal and $\dim H^{*}(G, k)/{\rm Res}_{G,E}^{-1}(\sqrt{0})=s$, we can conclude that

${\rm Res}_{G,E}^{-1}(\sqrt{0})=\mathfrak{p}$ and the first statement in the theorem follows.

We next shall prove the second statement in the theorem. Let $E\in \mathcal{A}_{s}$ and

assume

that the depth of $H^{*}(C_{G}(E), k)$ is $s$. Set $C=C_{G}(E),$ $N=N_{G}(E)$ and

we

shall

use

the

notations Section 2. As before let $\tau$ be

a

homogeneous element in $H^{*}(C, k)$ such that

${\rm Res}_{N,C}$Tr$C,N(\tau)$ is

a

regular element in $H^{*}(C, k)$ and restricts to

zero

on

every subgroup

$A$ of $C$ with $A\not\supset E$,

By Proposition 2.2, there exists a non zero homogeneous element $\eta_{0}\in$ Ker Res_$C,?t$

which is N-invariant. Notice that ${\rm Res}_{\overline{C}^{1}E}(\sqrt{0})\subset\sqrt{Ann_{H^{*}(Ck)}(\eta_{0})}$. Consider the element $\gamma=\eta_{0}\tau$

.

Then

an

entirely

same

argument

as

above,

we

have

${\rm Res}_{G,C}$ Tr$c,c(\gamma)={\rm Res}_{N,C}$Tr$C,N(\gamma)=\eta_{0}{\rm Res}_{N,C}$Tr$C,N(\tau)$

Set $\alpha=$ Tr$c,c(\gamma)\in H^{*}(G, k)$. We shall show that for $\rho\in H^{*}(G, k),$ $\rho\alpha=0$ if and only if ${\rm Res}_{G,C}(\rho)\eta_{0}=0$.

Assume that $\rho\alpha=0$. Then Tr$c,c({\rm Res}_{G,C}(\rho)\gamma)=0$. Hence ${\rm Res}_{G,C}$Tr$c,c({\rm Res}_{G,C}(\rho)\gamma)=$

$0$. By the similar argument in the above

we

have that

${\rm Res}_{G,C}$ Tr$c,c({\rm Res}_{G,C}(\rho)\gamma)={\rm Res}_{N,C}$Tr$c,N({\rm Res}_{G,C}(\rho)\gamma)=({\rm Res}_{G,C}(\rho)\eta_{0}){\rm Res}_{N,C}$Tr$C,N(\tau)$

As ${\rm Res}_{N,C}$Tr$C,N(\tau)$ is a regular element in _{$H^{*}(C, k)$}, it follows that ${\rm Res}_{G,C}(\rho)\eta_{0}=0$

.

Conversely, if ${\rm Res}_{G,C}(\rho)\eta_{0}=0$, then ${\rm Res}_{G,C}(\rho)\gamma=0$ and $\rho\alpha=$ Tr$C,G({\rm Res}_{G},c(\rho)\gamma)=0$.

A standard argument in commutative noetherian rings says that there exists $\delta\in$

$H^{*}(G, k)$ such that $\eta=\alpha\delta\neq 0$ and $\mathfrak{p}=Ann_{H^{*}(G,k)}(\eta)$ is

a

prime ideal. Then $\eta_{1}=$ ${\rm Res}_{G,C}(\delta)\eta_{0}\neq 0$ and by

an

entirely

same

argument

as

before, we have

Ann$H^{*}(G,k)(\eta)={\rm Res}_{G,C}^{-1}($Ann$H^{*}(C,k)(\eta_{1}))$

As ${\rm Res}_{\overline{C}^{1}E}(\sqrt{0})\subset\sqrt{Ann_{H^{*}(Ck)}(\eta_{0})}\subset\sqrt{Ann_{H^{*}(Ck)}(\eta_{1})},$ ${\rm Res}_{G,E}^{-1}(\sqrt{0})\subset\sqrt{Ann_{H^{*}(Gk)}(\eta)}=$

$\mathfrak{p}$ and $\dim H^{*}(G, k)/\mathfrak{p}=\dim H^{*}(H, k)/$ Ann$H^{*}(C,k)(\eta_{1})=s$. Thus _{${\rm Res}_{G,E}^{-1}(\sqrt{0})=\mathfrak{p}$} and

the second statement in the theorem follows.

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